What is the mass of water ejected per unit time from the nozzle in a toy rocket?

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The mass of water ejected per unit time from a toy rocket's nozzle can be calculated using the formula πr²pv, where p is the water's density, r is the nozzle's radius, and v is the water's speed. The calculation involves determining the volume of water expelled per second, which is derived from the nozzle's cross-sectional area multiplied by the velocity of the water. This volume is then converted to mass by multiplying it by the water's density. Understanding the geometry of the nozzle and the flow dynamics is crucial for this calculation. The discussion emphasizes the relationship between volume, density, and mass in fluid dynamics.
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Homework Statement


A toy rocket consists of a plastic bottle which is partially filled with water. The space above the water contains compressed air. At one instant during the flight of the rocket, water of density p is forced through the nozzle of radius r at speed v relative to the nozzle. Determine, in terms of p, r and v, the mass of water ejected per unit time from the nozzle


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The Attempt at a Solution


I know the answer is πr2pv. But I am not sure of exact explanation why this is the case.
 
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its not so much about momentum but really about understanding the geometry of the problem.

You know that the water has a certain velocity, you know how far a single particle of water will travell in one second. You are also given the radius of the nozzle therefore you can easily figure out the volume of the water that leaves the bottle every second.
It will be the area of the nozzle multiplied by the distance traveled by the water in one second ( that is V=pi*R^2*v)

once you have the volume its all easy as you can use the density to get the mass. it's just the volume multiplied by the density.
 
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